Optimal fourth- and eighth-order iterative methods for solving nonlinear equations with basins of attraction

Abstract

Nonlinear phenomena occur in diverse fields such as science, engineering and business. Research within computational science is continuously advancing, characterized by the development of new numerical techniques or the refinement of existing ones. However, these numerical techniques may be computationally expensive, while demonstrating superior convergence rate. By considering these demanding features, this paper aimed to devise new fourth- and eight-order iterative methods for root finding. This will be accomplished by taking the linear combination of Newton–Steffensen’s method and Yu and Xu’s method to obtain fourth-order method. We employed weight function approach to achieve eighth-order method. The proposed methods supports the Kung and Traub conjecture and hence are optimal by utilizing three function evaluations for fourth-order method and four functional evaluations for eighth-order method per cycle. The convergence criteria of the proposed schemes are thoroughly covered in the two primary theorems. To demonstrate the usefulness, validity and accuracy, we explore some real-world applications in civil and chemical engineering fields. In terms of the number of iterations, absolute residual errors, errors in consecutive iterations, the preassigned tolerance, convergence speed, percentage of convergent points, mean value of iterations for methods to converge and CPU time (sec), the numerical results obtained from the test examples illustrates that our proposed methods perform better than other methods of same order. Finally, several forms of complex functions are taken into consideration under basins of attraction in order to observe the overall fractal behavior of the proposed technique and some existing methods.